2.1.1 Encoder, laser modulator, optical channel

Figure 1: An optical communications system.

This paper concerns the communications system shown in Fig. 1. The information bits ${\bf U} = (U_1, \ldots, U_k)$ are i.i.d. binary random variables assumed to take on the values 0 and 1 with equal probability. The vector $\bf U$ is encoded to ${\bf X} = (X_1, \ldots, X_n)$, a vector of $n$ $M$-PPM symbols. Each $M$-PPM symbol is a number in $\{0, \ldots, M-1\}$ (or equivalently, a block of $\log_2 M$ bits, if $M$ is a power of two). There is one signaling slot, and $M-1$ nonsignaling slots for each $M$-PPM symbol. The symbol indicates to the modulator in which of the $M$ time slots of length $T_s$ to pulse the transmitting laser. Between each $M$-PPM symbol, the laser requires dead time $T_d$ to recharge and ready itself for sending another pulse. The laser is coupled to a telescope and pulses are transmitted through the optical channel to the receiving telescope, where background light also enters. In Fig. 1, the transmitting telescope, free space, background light, and receiving telescope are grouped under the term ``Optical Channel,'' whose output consists of photons that enter the detector.